I'd want *proof* (i.e., booting into different distros) that different
mkswap invocations generate different UUIDs.

I have no idea what proof I can provide except to say that I have been
caught by that.
I installed a second distribution and I could either not prevent it
from running mkswap or forgot to prevent it from doing so and I had,
when I rebooted into the first distribution, an error message about
swap. And it turned out that the swap UUID had been modified.

Expert install or simple install?
--
"History does not long entrust the care of freedom to the weak
or the timid." Dwight Eisenhower